Season chicken pieces generously with salt and pepper. Add oil to a large, heavy, oven-safe skillet and heat over high heat. When oil is almost smoking, carefully add chicken to pan and sear until almost cooked through; remove chicken from pan and set aside. Add shrimp to oil and sear until almost done; remove shrimp from pan and set aside.
Add more oil if needed, reduce the heat to medium, and then saute the onions and garlic until translucent, about 5 minutes; do not let onions and garlic brown. Add bell peppers and cook until softened, about 4 minutes. Add saffron and cook until fragrant, about 1 minute. Add chorizo and brown for 2 minutes. Add tomato paste and cook for about 5 minutes. Add rice, and stir until the rice is well coated, about 1 minute. Add stock, return the chicken and shrimp to the pan, and add mussels and peas.
Transfer skillet to the oven and bake for 25 to 30 minutes. Remove from oven and drizzle vinaigrette over. Refrigerate and serve chilled or at room temperature. If serving at room temperature, refrigerate until 30 minutes before serving and then bring to room temperature.
Fresh Herb Vinaigrette:
Put chopped herbs in a medium bowl. Add garlic and vinegar. Season, to taste, with salt and pepper. Whisk in olive oil until combined. Set aside until paella is ready.